AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 86:537-547 (1991)
Body Size, Locomotion, and Long Bone Cross-Sectional Geometry in lndriid Primates B DEMES, W L JUNGERS, AND K SELPIEN Department of Anatomical Sciences, State University of New York, Stony Brook, New York 11794 (B D , W L J I and Abteilung Funktionelle Morphologie, Ruhr Uniuersitat, 4630 Bochum, Germany ( K S )
KEY WORDS
Indriids, Long bone geometry, Scaling
ABSTRACT
The geometry of the midshaft cross-sections of the femur and humerus of five indriid species was analysed. Internal (marrow cavity) and external diameters were measured on X-rays in the anteroposterior (a-p) and mediolateral (m-1)planes; cross-sectional areas, second moments of area, and section moduli were calculated using formulae for a hollow ellipse. Cortical thickness, robusticity indices (relating external diameters to the length of the bones), and a-p/m-1 shape variables were also calculated. Model I1 regression was supplemented by analyses of correlation between size and shape. Indriids are saltatory, i.e., their locomotion is dominated by the hind limbs. Accordingly, the femur is more rigid than the humerus, and it shows a consistent difference between the a-p and m-1 planes in measures related to bending strength. Cortical thickness varies considerably both within and across species. The type specimen of the new species Propithecus tattersalli is virtually indistinguishable from P. uerrealwci on the basis of its long bone cross-sectional geometry. Femoral robusticity is uncorrelated with size, but humeral robusticity decreases significantly with increasing size. Femoral shape variables (a-p/m-1) are all negatively correlated with body size, indicating that m-1 dimensions of the femur increase at a faster rate than do a-p dimensions. The highly loaded plane of movement seems to be more reinforced in the smaller species. Contrary to static biomechanical scaling predictions of positive allometry, all cross-sectional parameters scale relatively close to isometry. It is concluded that either changes in locomotor performance must compensate for the weight-related increase in forces and moments or that the larger-bodied animals operate appreciably closer to the limits of their safety margins.
The magnitude and direction of external forces acting on a n animal's bones are determined by its positional repertoire and its weight.' The cross-sectional dimensions of the bones largely determine their ability to withstand the stresses created by these forces. High correlations between locomotion, body size, and cross-sectional dimensions of long bones can therefore be expected. In acknowledging this causal association, an increasing number of studies have collected valuable information on the geometry of long bone cross sections. For various pri'Weight and mass are related by the gravitational constant of 9.81 and may be used interchangeably for animals moving under the influence of earths gravity.
0 1991 WILEY-LISS,INC.
mate species a correlation has been established between the ratio of forelimb to hind limb bone rigidity and the role of the limbs in support and propulsion (Schaffler et al., 1985; Ruff, 1987a; Demes and Jungers, 19891, between cross-sectional shape and locomotor activity (Preuschoft, 1969; Jungers and Minns, 1979; Burr et al., 1982, 1989; Ruff, 1987b, 1989; Demes and Jungers, 1989), and between body weight and crosssectional dimensions, allowing the prediction of weight from cross-sectional data in fossils (Ruff, 1987a, 1989; Ruff et al., 1989). In this study we extend this biomechanical Received December 12,1990; accepted June 19,1991
538
B. DEMES ET AL.
approach to the analysis of skeletal design in to a broad sampling strategy that combines a group of closely related, saltatory pri- animals of different locomotor mode and mates. Leapers have a clear differentiation with different phylogenetic affinities. In a n between the role of the forelimb and hind analysis limited to artiodactyls, but ranging limb in locomotion a s well as a preferentially over three orders of magnitude in body mass, loaded plane for the propulsive hind limb; Selker and Carter (1989) report that crossaccordingly, they should exhibit a pro- sectional parameters scale in a positively nounced functional differentiation in cross- allometric fashion. sectional dimensions. For the present analysis we have deterA related and much debated issue con- mined the midshaft cross-sectional dimencerns the scaling of long bone dimensions sions of the humerus and femur in a group of with body weight. Mechanically inspired closely related prosimian primates in the scaling models predict relative increases family Indriidae.2 The difference in body (positive allometry) in external diameters of mass between the smallest and the largest long bones with increasing body weight. Ga- species in our sample is approximately a lilei (1637) suggested this was due to weight factor of 7 to 8 (see Table 1 and below). In increasing in proportion to volume, and bone analysing closely related species that also cross section increasing in proportion only to share the same general locomotor mode, difarea. Whereas Galilei’s model assumes (im- ferences due to phylogenetic “noise” and plicitly) that axial compression is the pre- variation in locomotor category are reduced dominant loading regime, more recent mod- greatly, thereby revealing size effects more els assume either Euler buckling (McMahon, clearly. This group corresponds closely to 1973; Alexander, 1988) or bending (Kum- what McMahon (1984) h a s recently demer, 1975; Prange, 1977) to be the critical scribed a s the most appropriate type of samloading regimes of long bones. Depending on ple for testing mechanical models of long the model, the expected scaling coefficients bone scaling. By analysing cross-sectional vary, but they all predict values greater than dimensions in addition to external diameisometry or geometrical scaling; i.e., they ters, a much more direct measure of the predict that larger animals should have rel- mechanical strength of a bone is obtained. The five indriid species included in our atively thicker, more robust bones. These scaling models are static in nature and ne- sample are all specialized “vertical clingers glect to account for possible changes in the and leapers” (Napier and Walker, 1967). loading patterns due to differences in loco- They prefer vertical tree trunks for takeoffs motor performance, external forces and their and landings, and their positional repertoire lever lengths, and muscle forces and their is characterized by vertical clinging poslever lengths. Such changes may occur as tures. Quantitative data on locomotor becorrelates of body size (Biewener, 1982, havior are available for sifakas (genus Pro1983,1990; Rubin and Lanyon, 1984; Alex- pithecus) only. About 50% of all locomotor ander, 1985,1989). and postural bouts is leaping, followed by Positive allometry of long bone diameters climbing and clinging (Gebo, 1987; Dagosto, was documented for bovids (McMahon, 1975; 1989).All these activities are hind-limb domAlexander, 1977; Scott, 1985) and for run- inated, which means that the greater share ning birds (Maloiy et al., 1979) and was also of total body weight is supported and accelfound in mixed samples of primate species erated by the hind limbs. Sifakas can easily (Stahl and Gumerson, 1967; Ruff, 1987a). In cover a horizontal distance of 10 meters in a analysing a wide range of different animals single leap (Petter, 1962). The locomotion of from shrews to elephant, Alexander et al. the other species of the family is reported to (1979) and Biewener (1983) found scaling be similar to that of the sifakas (Petter, 1962; coefficients for external diameters closer to Petter et al., 1977; Tattersall, 1982). isometry, but the cross-sectional areas and MATERIALS AND METHODS area moments scale slightly positively alloCross-sectional dimensions were determetrically in ten species of mammals from mouse to horse (Biewener, 1982). Positive mined from measurements taken on radioallometry of cross-sectional areas and area graphs, and a n elliptical model was emmoments was also found in two taxonomically mixed primate samples (Schaffler 2Jenkins (1987) has argued recently that the proper familial et al., 1985; Ruff, 1987a).The interpretation designation for this group is Indridae. We have opted for the “conventional” spelling. of these results is complicated somewhat due
539
INDRIID LONG BONE GEOMETRY
TABLE I. Sample composition Origin
Zndri indri Propithecus diadema P. uerreauxi P. tattersalli Auahi laniger
N
Wild
14/15 10 16/18 1 11/12
7/8 8 9 1 8/9
zoo
?
Body mass
2
7 2 5/7
-
3
7-5 kg (1) 6.2 k g (2,3) 3.7 k g (4,5) 3.2 k g (5) 1.0 kg (1,3,5)
?indicates no locality information; Ns = femur/humerus. if different in numbers. Body weights are basedonthefollowingsources:1:Stephan and Bauchot(l965)report a body weightof6.25 kgfor a wild-shothdrifemale. Judging from its head and body length (Rehkamper, personal communication),this animal is smaller than the species mean (Tattersall, 1985). We have thereforeinserted a higher valueandin addition calculate regressions with two body weights for Zndri;2:Smithsonian Field Collection; 3: Glander et al., 1990; 4: Tattersall, 1985; 5: Duke Primate Center records, Kappler, 1990.
ployed in all calculations. Choice of this simple model seems reasonable because the cross-sectional perimeters of the indriid bones are quite regular, lacking crests and other local protuberances in the midshaft region. For a sample of human humeral sections, Fresia et al. (1990) have recently demonstrated a close correlation between the cross-sectional geometrical properties calculated from measurements taken on X-rays in two planes (using a n elliptical model) and determined directly, lending support to the utility of our approach. The midshaft crosssection was chosen as the focal point of our analyses because there are empirical data indicating that it suffers the highest bending stresses (Biewener and Taylor, 1986). Humeri and femora of five indriid species were X-rayed from the following collections: American Museum of Natural History (New York), Museum National d’Histoire Naturelle (Paris), Anthropologisches Institut und Museum (Zurich), Cleveland Museum of Natural History (Ohio),Museum of Comparative Zoology (Boston), and the Rijksmuseum van Natuurlijke Historie (Leiden). The sample composition is provided in Table 1. Samples include several zoo specimens and animals without precise locality data in order to maximize sample sizes of these relatively rare species. The new species Propithecus tattersalli (Simons, 1988) is represented by the type specimen, so far the only skeleton available for study. All specimens were adult a s judged from their external appearance (i.e., epiphyseal lines unvisible). Individual body masses were not available for the majority of specimens. For the analysis of sizerelated differences, a n average body mass for each species was used (Table 1).Sexes were pooled as all five species are nondimorphic with respect to their body size (Kappler,
1990; Glander et al., 1990). For P. tattersalli, the mass of the type specimen was known from the records of the Duke University Primate Center. The functional length of the bones was measured directly as the greatest distance between proximal and distal joint surfaces. This length measurement may deviate slightly from the greatest length in the case of the femur, where the tip of the greater trochanter can be the most proximal extension. External and internal diameters a t midshaft were measured on the X-rays with a magnifying lens ( x 8 ) with a scale. These measurements were taken in the anteroposterior (a-p) and the mediolateral (m-1) planes, with the planes being determined by the orientation of the distal joint. The a-p plane thus coincides with the plane of movement a t the knee or the elbow joint, respectively, with the m-1 plane being perpendicular to it. The reliability ofthe method was tested for a subset of data by comparing the measurements of external diameters taken from the X-rays with measurements taken on the bone directly with digital calipers. The percentage deviation of the X-ray measurements from the bone measurements was 2% on average. Trotter and Peterson (1967), by comparison, found a 5% deviation for the transverse diameter of the human femur when comparing X-ray with direct bone measurements. Cross-sectional areas (A), a s a measure related to the compressive strength of the bone, and second moments of area (I) as well a s section moduli (21, as reflections of its bending strength, were estimated using formulae for a hollow ellipse (Fig. 1;Roark and Young, 1976):
540
B. DEMES ET AL.
A
= TT
(ab
-
aibJ mm2
L - p - d 4 (a3b - aI3bb,) mm4 Im., = T T / ~(ab3 - a1b,3)mm4
Za.p = IJa mm3 Zm., = ImJb mm3 Ia.p is proportional to the bending rigidity of the cross section against bending moments in the sagittal plane (around a transverse axis); I,,., to the rigidity against bending moments in the frontal plane (around a sagittal axis). The section moduli are a more direct measure of bending strength, because they compensate for the distance from the neutral axis to the bone’s outer perimeter (where the highest bending stresses occur). Both I and Z are reported here because comparative data on the bending strength of long bones are currently available primarily as area moments. The equations used above account for differences in diameters as well as in wall thicknesses between the two planes, but not in wall thicknesses within a plane. As opposite cortices are similar in thickness in our sample, the potential error introduced by this is minor (see Biknevicius and Ruff, in press, for the asymmetrical ellipse model calculations). As a measure of cortical thickness, K-values were determined as internal (marrow cavity) diameter as a ratio of external diameter (Currey and Alexander, 1985). The robusticity index was calculated as the ratio of average external midshaft diameter and bone length. Ratios of diameters, area moments, and section moduli in the two planes (a-p and m-1) were also calculated a s indicators of midshaft cross-sectional shape. The measured and derived variables were compared within each bone for the two planes (a-p and m-11, and within each species for forelimb and hind limb bones (femur and humerus) using the Mann-Whitney U-test; a nonparametric test was employed because variables tended to be nonnormally distributed. The scaling relationship of variables to body size (average body mass) was explored via reduced major axis regressions of logarithmically transformed data (Ricker, 1984; Rayner, 1985). Scaling was also investigated via correlation between size and shape variables (Mosimann and James, 1979). Spearman rank order correlations were calculated between body size and the robusticity indices and a-p/m-1 indices. No significant correlation here implies “isometry”; allometry is
antero posterior
-
mediolateral
Fig. 1. Ellipse approximating cross-sectional areas from anteroposterior and inediolateral diameters. d a , = ?/A externayinternal a-p diameter; b/b, = ‘/z externayinternal m-1 diameter.
indicated simply by significant positive or negative correlations. RESULTS
Variability The coefficients of variation for the linear measurements on the external surface of the bones (length, external midshaft diameters) range between 4.7 and 10.4. This is slightly higher than the variability found in external bone measurements of mammals in general (Yablokov, 1974).This may be due to the fact that some of the species (especially P. uerreauxi and P. diadema) in our sample combine several subspecies with a relatively wide range of body sizes (Tattersall, 1982). For the internal diameter (width of marrow cavity) the coefficients of variation are even higher (6.9-17.61, which is probably due to a somewhat higher measuring error, related to its absolutely smaller dimension and the less clearly demarcated border of the internal surface of the cortical wall. The average coefficient of variation for the cross-sectional areas is 16.1(range: 9.9-24.6), for the section moduli 20.9 (range: 13.5-29.7) and for the second moments of area 28.5 (range: 17.9-
2. p, 2, I = section moduli; I.
Zndri indri femur humerus Propithecus diadema femur humerus Propithecus verreauxi femur humerus Propithecus tattersalli femur humerus Avahi laniger femur humerus 8.7 7.0
9.9 7.8 6.4 f 0.4 4.7 f 0.5
184.9 100.2
128.4 f 7.6 62.3 f 3.3
I, I = second moments of area.
*
6.0 f 0.3 4.9 0.3
+
19.9 i 2.8 9.5 1.5
45.2 21.0
42.8 i 10.5 20.9 f 4.4
9.1 0.9 6.9 i 0.7
9.6 i 0.6 7.2 f 0.7
181.3 f 9.0 97.5 f 6.7
*
69.9 f 6.9 33.2 i 4.2
10.8 f 0.6 8.5 f 0.4
11.3 f 0.7 8.5 f 0.8
209.2 11.4 120.6 f 7.0
+
77.9 f 9.6 36.9 f 6.2
11.7 f 0.7 9.2 f 0.8
Area mm2
+
m-l diam. mm
12.1 f 0.7 9.2 0.8
a-p diam. mm
of
240.2 f 11.7 133.0 f 6.3
mm
Length
TABLE 2. External dimensions and cross-sectional parameters
21.6 f 3.7 8.3 f 2.1
74.8 30.0
71.8 i 17.9 27.4 f 8.1
127.1 f 19.8 50.0 i 9.8
152.7 f 23.0 62.0 f 14.3
mm3
&.&I
L-p
70.0 f 15.7 20.2 f 6.9
19.8 3.1 8.8 k 2.0
+
370.4 117.1
350.1 f 111.6 101.0 f 40.0
67.1 i 19.6 27.2 f 8.0 65.6 28.5
726.3 f 161.8 217.3 f 61.7
928.8 f 188.0 290.3 f 88.9
mm4
119.0 f 16.1 50.1 f 6.8
147.6 f 24.2 61.8 f 14.7
mm3
Zm.1
limb bones (means and one standard deviation) Im-I
59.8 5 12.3 22.0 f 6.7
283.5 99.8
313.0 i 127.1 96.6 f 40.6
+
646.0 f 116.7 213.3 38.3
868.4 k 195.9 290.1 f 93.9
mm4
542
B. DEMES ET AL
tion of two K-values in the mediolateral plane (P. diadema: P < 0.05; P. uerreauxi: n.s.1. Relative to its length the humerus is, however, the more robust bone (Table 4).
T A B L E 3. Ratios of internal to external midshaft diameters (K-values) (means and one standard deviation)
Indri indri femur humerus Propithecus diadema femur humerus Propithecus uerreauxi femur humerus Propithecus tattersalli femur humerus Auahi laniger femur humerus K
=
K-value a-p
K-value m-l
0.55 f 0.03 0.66 f 0.05
0.54 f 0.05 0.67 f 0.05
0.49 k 0.06 0.65 f 0.05
0.54 f 0.08 0.63 f 0.06
0.60 f 0.07 0.70 f 0.04
0.63 f 0.07 0.66 f 0.04
0.57 0.74
0.59 0.68
0.56 f 0.05 0.70 f 0.04
0.60 f 0.04 0.68 f 0.04
internal diametedexternal diameter
42.1). In much larger samples of hominoid primates, Ruff (1987a) reports similarly high coefficients of variation (7-26 for crosssectional areas, and 14-47 for polar moments of inertia). These data demonstrate that this degree of variability is not due to small sample sizes; rather, it is simply the consequence of higher than linear dimensionality such a s mm2 and mm4 (Lande, 1977).
Femur us. humerus The femur has greater outer diameters, wall thicknesses, areas, second moments of area, and section moduli than does the humerus (Tables 2, 3). All differences are significant a t the P < 0.01 level, with the excep-
a-p us. m-1plane In the femur, there is a consistent difference between the a-p and the m-1 plane such that the a-p diameters, cortical thicknesses, second moments of area, and section moduli are greater than the corresponding m-1 parameters (with one exception of wall thickness in Indri; Tables 2 4 ) . However, none of these differences is statistically significant a t the P < 0.01 level. There are no consistent differences between the two planes in the humerus; i.e., a-p and m-1 variables are very similar (Tables 2 4 ) . K-values The wall thicknesses vary considerably, within a cross section between the a-p and m-1 plane, between the femur and humerus of the same species, and interspecifically. Figure 2 illustrates this variation for the femur by contrasting a n a-p/m-1plot of K-values with one for section moduli. The femora tend to have lower K-values (i.e., thicker cortical walls) than do the humeri (Table 3). The variation in K does not show any obvious correlation with size, taxonomy, or with the two planes of a bone. Size All variables are highly correlated with body mass (Table 5). With the exception of bone length (cf. Jungers, 19851, most scaling coefficients are quite close to isometry: for
T A B L E 4. Robusticity and diaphyseal shape of limb bones (means and one standard deviation)
Zndri indri femur humerus Propithecus diadema femur humerus Propithecus uerreauxi femur humerus Propithecus tattersalli femur humerus Auahi laniger femur humerus Robusticity index = (a-p diameter
Robusticity index
a-p diam. m-1 diam.
1a.p
2a.p
1m.I
Zm.1
9.9 f 0.7 13.9 f 1.0
1.039 i 0.070 1.002 f 0.084
1.085 i 0.149 1.010 f 0.163
1.040 f 0.080 1.004 f 0.071
10.6 i 0.7 14.1 f 0.9
1.049 +_ 0.052 1.008 f 0.076
1.123 i0.119 1.006 -+ 0.164
1.067 k 0.064 0.992 f 0.095
10.3 f 0.7 14.5 f 0.9
1.062 f 0.069 1.042 f 0.087
1.151 f 0.149 1.055 f 0.159
1.080 f 0.073 1.007 f 0.070
10.1 14.8
1.140 1.013
1.306 1.173
1.140 1.053
9.7 f 0.5 15.5 f 1.3
1.074 f 0.055 0.964 f 0.054
1.175 f 0.130 0.914 0.108
1.091 f 0.067 0.945 f 0.060
+ m-l diameter) X 100Aength. Ia.p, I,
1=
*
second moments of area; Z.
p,
Z,.
= section moduli.
543
INDRIID LONG BONE GEOMETRY
O8
r250 200
l------
E
0.5
T Avahi loniger
V Propithecus verreaux 0
P. tattersaiii
0 P. diaderna 0 indri indri
0.4 0.4
0.5
k-value
0.8
0.7
0.6 0-D
femur
0
50
100 0-p
150
:
200
section mod. f e m u r
Fig. 2. Femur m-1 parameters plotted against corresponding a-p parameters. a: K-values; b: section moduli.
TABLE 5. Reduced major axis regressions of variables against body mass (log-log transformed data) Bone length femur humerus Diameter a-p femur humerus Diameter m-1 femur humerus Cross-sectional area femur humerus Section modulus a-p femur humerus Section modulus m-1 femur humerus Area moment a-p femur humerus Area moment m-1 femur humerus
Correlation
y-intercept
Slope
0.983 0.993
4.844 4.130
0.301 (0.290) 0.377 (0.363)
0.992 0.978
1.868 1.569
0.319 (0.307) 0.336 (0.323)
0.998 0.991
1.781 1.574
0.335 (0.322) 0.314 (0.302)
0.992 0.992
2.964 2.225
0.696 (0.669) 0.692 (0.665)
0.996 0.993
3.075 2.117
0.987 (0.949) 1.008 (0.969)
0.997 0.994
2.971 2.149
1.009 (0.970) 0.981 (0.943)
0.995 0.992
4.260 3.016
1.305 (1.255) 1.339 (1.287)
0.998 0.994
4.066 3.046
1.344 (1.292) 1.292 (1.242)
Correlations, intercepts, and slopes are given forregressions calculated with a body mass of 7 kgfor Indri.They do not changemuch if Indri is inserted with a body mass of 8 kg (slopes in parentheses; see Methods section).
the linear dimensions (diameters) approximately 0.33 W31, for areas approximately 0.67 (2/3), for section moduli approximately 1 ( 3 / 3 ) , and for area moments approximately 1.33 (Y3). Femoral robusticity follows this isometry trend in that this index is uncorrelated with body mass (Table 6, Fig. 3). Hum-
era1 robusticity, however, decreases significantly with increasing body mass (Table 6). Zndri has the least robust humerus and Avahi has the most robust (Table 4, Fig. 3); presumably, the positive allometry of humerus length contributes to this trend (note that these data modify the results reported
544
B. DEMES ET AL.
TABLE 6. Rank-order correlations between body mass and diaphyseal shape variables Variable Humerus robusticity index a-p/m-l diameter Ia-p/Im-l
Za.p/Zm.l
Femur robusticity index a-p/m-1 diameter Ia.p~Im.l Z,./Z, I
Spearman correlation
Interpretation
-1.000* 0.100 0.100 0.100
neg. allometry isometry isometry isometry
0.400 -0.900* -0.900* -0.900*
isometry neg. allometry neg. allometry nee. allometrv
*Indicates significance (P< 0.05); negative allometry implies that index decreases; isometry implies no significant change in index with increasing size (geometric similarity preserved).
by Jungers, 1985, for the humerus of indriids). Significant deviations from isometry are also reflected in shape changes of the femoral midshaft cross-section with increasing size. The a-p to m-1 ratios of diameters, second moments of area, and section moduli decrease with increasing size (Table 61, indicating that the difference in strength between these two planes is less pronounced in the larger-bodied species. These significant size-related shape changes are the combined effects of the slightly divergent a-p and m-1 scaling trends seen in Table 5. DISCUSSION
Our data corroborate hypotheses about the functional differentiation of long bones according to habitual positional behaviors. The bone of the propulsive hind limb (femur) that has to withstand the higher forces is more resistant to loads than is the bone ofthe corresponding segment of the forelimb (humerus). The femur also offers the greater resistance in the plane of movement (a-p), where it is probably exposed to higher bending moments than in the plane perpendicular to it (m-1). This difference is, however, quite subtle, and changes with size, so that inferences derived solely from cross-sectional shape about the likely mode of locomotion in fossils should be stated carefully. It should also be mentioned in this context that there are examples in which the long axis of the elliptical cross section of a bone is not in the plane of major bending (Lanyon and Rubin, 1985). In some experiments changing or increasing the loading of bones, on the other hand, bone apposition and mechanical reinforcement is found in the loaded plane
(e.g., Liskova & Helt, 1971; Gordon, et al., 1989). The external measurements distinguish the two bones and the two planes within a bone in much the same way a s do the crosssectional parameters. They accurately predict aspects of internal geometry of the bones and can therefore be taken a s indirect indicators of their mechanical strength when cross-sectional data cannot be obtained (Ruff, 1987b). A close correlation between external bone diameters and second moments of cross-sectional area was also documented by Jungers and Minns (1979) for a mixed primate sample. This is not totally unexpected as the outer parts of cross-sectional area contribute preferentially to bending strength (compare equations for I and Z in the Methods section). The one specimen of the new species P. tattersalli, although weighing slightly less than a n average P. uerreauxi, falls in the upper range of verreauxi’s spectrum with regard to its humeral and femoral dimensions (Tables 2-4). The type specimen is a very robust male. The variation in cortical thickness does not exhibit a very coherent pattern. K-values for the femur vary around an optimal value of 0.55, predicted by Currey and Alexander (1985) to achieve maximum ultimate bending strength with a minimum of bone mass. K-values for the humerus lie closer to what is the assumed optimum for yield bending strength (0.65:Pauwels, 1974; 0.67: Currey and Alexander, 1985). However, both femoral and humeral K-values vary considerably. These data, as well as similarly variable data on lorisine long bones (Demes and Jungers, 19891, suggest that achieving a maximum bending strength with a minimum of bone mass is probably not the only factor determining the wall thickness of long bones. The virtual isometry of all cross-sectional parameters with size clearly contradicts all competing static scaling predictions of positive allometry (Galilei, 1637; McMahon, 1973; Kummer, 1975; Prange, 1977; Alexander, 1988). The findings also suggest that Biewener’s (1990) size categories (e.g., isometry limited to mammals between 0.001-0.1 kg) are not hard and fast. In addition, they stress the importance of multifactorial approaches toward bone scaling, taking into account changes in locomotor performance and external forces with size (Biewener, 1983, 1990; Rubin and Lanyon, 1984: dynamic strain similarity; Alexander, 1989).
INDRIID LONG BONE GEOMETRY
545
G
a U
Fig. 3. Femur and humerus of Auahi laniger (right) and Indri indri (left)drawn at the same length (scale bars indicate 1 cm). Notice the similar femur proportions (a)and the increased humeral robusticity of the smaller-bodied species (b).
Options to cope with increasing weight-related forces include changes in locomotor behavior that reduce the external loads (e.g., reduce speed; Alexander, 1989) and their moment arms (e.g., straighter legs, shorter strides; Alexander, 1989; Biewener, 1989, 1990; Bertram and Biewener, 1990), and changes in body proportions (e.g., reduce bone length and bony moment arms, increase muscle moment arms; Biewener, 1990). For example, leaping galagos and L. catta show distinct, size-related differences in the way they accelerate for takeoff. The small-bodied species exert relatively higher forces and assume smaller angles a t the joints of the accelerating hind limb than do the larger-bodied species (Gunther, 1989; Demes and Gunther, 1989). Although corresponding data for the indriids are not yet available, it is not unlikely that they compensate in a similar way for increasing weight-related forces. The decreasing a-p to m-1 ratios in variables of the femur cross section related to bending strength support this view. Alternatively, it has to be assumed
that the bones of the larger-bodied animals suffer from greater stresses and operate closer to their safety margins (Biewener, 1982). In vivo strain gage data, however, indicate, that bone strains, although variable, are in the same order of magnitude in animals over a wide range of body sizes (Rubin and Lanyon, 1984). When comparing our results with those of other studies of long bone scaling, it is obvious that animals follow different strategies to maintain strain similarity with increasing size (although dynamic similarity seems to apply a t least to a certain degree: Alexander, 1985). In other analyses of closely related species that also share the same mode of locomotion, slight to moderate positive allometry was found for most of the crosssectional parameters of artiodactyl long bones and for “pongid” tibiae and femora (Ruff, 1987a1, and strong positive allometry for lorisine long bones (Demes and Jungers, 1989). I n lorises, the increase in bone dimensions with increasing size is even more pronounced than could be expected from docu-
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B. DEMES ET AL
mented changes in kinematic and dynamic parameters with increasing size (Demes et al., 1990), raising the likelihood of greater safety factors for the larger-bodied species as a n adaptation to infrequent but critical stresses (Alexander, 1984). Indriids appear to have “opted” for a rather different solution given the virtual isometry in their long bone cross-sectional dimensions. We still have much to learn about the interactions in body size, skeletal design, and animal behavior, but we believe that the analytical strategy used here that focuses on closely related animals of different size but similar design has much to offer. ACKNOWLEDGMENTS
We would like to thank the curators of the collections listed under “Methods” for giving us access to specimens in their care. Special thanks for help with the radiographs go to A. Biknevicius and N. Feinberg (New York), F.A. Jenkins (Harvard), F.K. Jouffroy (Paris), B. Latimer (Cleveland), P. Schmid (Zurich), C. Smeenk and S. Zeilstra (Leiden). Unpublished body size data were kindly provided to us by B. Colfman (Duke University Primate Center) and G. Rehkamper (Cologne). This research was supported in part by DFG grant De 390/1-1 and NSF BNS 8606781. LITERATURE CITED Alexander RMcN (1977) Allometry of the limbs of antelopes (Bouidae).J. Zool., Lond. 183t125-146. Alexander RMcN (1984) Optimum strength for banes liable to fatigue and accidental fracture. J. Theor. Biol. 109:621-636. Alexander RMcN (1985) Body support, scaling, and allometry. In M Hildebrand, DM Bramble, KF Liem, and DB Wake (eds.):Functional Vertebrate Morphology. Cambridge, Mass.: The Belknap Press, pp. 26-37. Alexander RMcN (1988) Elastic Mechanisms in Animal Movement. Cambridge: Cambridge University Press. Alexander RMcN (1989)Mechanics of fossil vertebrates. J . Geol. SOC., Lond. 146:41-52. Alexander RMcN, Jayes AS, Maloiy GMO, and Wathuta EM (1979) Allometry of the limb bones of mammals from shrews (Sorex)to elephant (Loxodonta).J. Zool., Land. 189:305-3 14. Bertram JEA and Biewener AA (1990)Differential scaling of the long bones in the terrestrial Carnivora and other mammals. J. Morphol. 204t157-169. Biewener AA (1982) Bone strength in small mammals and bipedal birds: Do safety factors change with body size? J. Exp. Biol. 98t289-301. Biewener AA (1983) Allometry of quadrupedal locomotion: The scaling of duty factor, bone curvature and limb orientation to body size. J. Exp. Biol. 105t147171. Biewener AA (1989) Mammalian terrestrial locomotion and size. BioScience 139:776-783.
Biewener AA (1990)Biomechanics of mammalian terrestrial locomotion. Science 250:1097-1103. Biewener AA and Taylor CR (1986) Bone strain: A determinant of gait and speed? J. Exp. Biol. 123:383400. Biknevicius AR and Ruff CB (in press) Use of biplanar radiographs for estimating cross-sectional geometric properties of mandibles. Anat. Rec. Burr DB, Piotrowski G, Martin BR, and Cook PN (1982) Femoral mechanics in the lesser bushbaby (Galago senegalensis): Structural adaptations to leaping in primates. Anat. Rec. 202:419429. Burr DB, Ruff CB, and Johnson C (1989) Structural adaptation of the femur and humerus to arboreal and terrestrial environment in three species of macaque. Am. J. Phys. Anthropol. 79t357-367. Currey J D and Alexander RMcN (1985)The thickness of the walls oftubular bones. J. Zool., Lond. 206:453-468. Dagosto M (1989) Locomotion of Propithecus diadema and Vurecia uarzegata at Ranomafana National Park, Madagascar. Am. J. Phys. Anthropol. 78t209. Demes B and Gunther MM (1989) Biomechanics and allometric scaling in primate locomotion and morphology. Folia Primatol. (Basel)53t125-141. Demes B and Jungers WL (1989)Functional differentiation of long bones in lorises. Folia Primatol. (Basel) 52t58-69. Demes B, Jungers WL, and Nieschalk U (1990)Size- and speed-related aspects of quadrupedal walking in slender and slow lorises. In FK Jouffroy, MH Stack, and C Niemitz (eds.): Gravity, Posture and Locomotion in Primates. Florence: I1 Sedicesimo, pp. 175-197. Fresia AE, Ruff CB, and Larsen CS (1990) Temporal decline in bilateral asymmetry of the upper limb on the Georgia coast. In CS Larsen (ed.):The Archaeology of Mission Santa Catalina de Guale: 2. Biocultural Interpretations of a Population in Transition. New York: Anthropological Papers of the American Museum of Natural History 68:121-150. Galilei G (1637)Discorsi e Dimostrazioni Mathematiche Intorno a Due Nuove Scienze. New, translated edition (1914).Dialogues Concerning two New Sciences. New York: MacMillan. Gebo DL (1987) Locomotor diversity in prosimian primates. Am. J. Primatol. 13:271-281. Glander KE, Wright PC, and Daniels PS (1990)Morphometrics of six rain forest prosimian species from Southeastern Madagascar. Am. J. Primatol. 81229. Gordon KR, Per1 M, and Levy C (1989) Structural alterations and breaking strength of mouse femora exposed to three activity regimens. Bone 10:303-312. Gunther MM (1989)Funktionsmorphologische Untersuchungen zum Sprungverhalten an mehreren Halbaffenarten. Berlin: Ph.D. thesis. Jenkins PD (1987) Catalogue of Primates in the British Museum of Natural History. Part IV. Suborder Strepsirrhini. London: British Museum. Jungers WL (1985)Body size and scaling of limb proportions in primates. In WL Jungers (ed.): Size and Scaling in Primate Biology. New York: Plenum Press, pp. 345-381. Jungers WL and Minns R J (1979) Computed tomography and biomechanical analysis of fossil long bones. Am. J. Phys. Anthropol. 50:285-290. Kappler PM (1990) The evolution of sexual dimorphism in prosimian primates. Am. J. Primatol. 21:201-214. Kummer B (1975) Grundsatzliche Bemerkungen zum Einfluss der Korpergrosse und der Gravitation auf die Konstruktion des Bewegungsapparates landbe-
INDRIID LONG BONE GEOMETRY wohnender Tetrapoden. Aufsatze u. Reden Senckenb. naturf. Ges. 27:69-84. Lande R (1977) On comparing coefficients of variation. Syst. Zool. 26:214-217. Lanyon LE and Rubin CT (1985) Functional adaptation in skeletal structures. In M Hildebrand, DM Bramble, KF Liem, and DB Wake (eds.): Functional Vertebrate Morphology. Cambridge, Mass.: The Belknap Press, pp. 1-25. Liskova M and Heit J (1971)Reaction of bone to mechanical stimuli. Part 2. Periosteal and endosteal reaction of tibia1 diaphysis in rabbit to intermittent loading. Folia Morphol. (Warsz) 19:301-317. Maloiv GMO. Alexander RMcN. Niau R. and Javes AS (19f9)Allometry of the legs of rtnning birds. i.Zool., Lond. 187:161-167. McMahon TA (1973) Size and shape in biology. Science 179:1201-1204. McMahon TA (1975) Allometry and biomechanics: Limb bones in adult ungulates. Am. Natur. 109:547-563. McMahon TA (1984)Muscles, Reflexes, and Locomotion. Princeton: Princeton University Press. Mosimann J E and James FC (1979) New statistical methods for allometry with application to Florida red-winged blackbirds. Evolution 33t444459. Napier JR and Walker AC (1967) Vertical clinging and leaping-a newly recognized category of locomotor behaviour of primates. Folia Primatol. (Basel) 6204219. Pauwels F (1974)Uber die Bedeutungder Markhohle fur die mechanische Beanspruchung des Rohrenknochens. Z. Anat. Entwick1.-Gesch. 145t81-85. Petter JJ (1962) Recherches sur l’l&ologie et 1’Ethologie des Lemuriens Malgaches. Mem. Mus. Nat. dHist. Nat. 27(1). Petter JJ, Albignac R, and Rumpler Y (1977) Mammiferes lemuriens (Primates, Prosimiens). Faune Madagascar 44: 1-513. Prange HD (1977) The scaling and mechanics of artbropod exoskeletons. In J T Pedley (ed.): Scale Effects in Animal Locomotion. London: Academic Press, pp. 169-1 81. Preuschoft H (1969) Statische Untersuchungen am Fuss der Primaten. I. Statik der Zehen und des Mittelfusses. Z. Anat. Entw. Gesch. 129:285-345. Rayner JMV (1985) Linear relations in biomechanics: The statistics of scaling functions. J. Zool., Lond. 206:415439. Ricker WE (1984)Computation and uses ofcentral trend lines. Can. J. Zool.62t1897-1905.
547
Roark R J and Young WC (1976)Formulas for Stress and Strain, 5th Edition. Auckland: McGraw-Hill. Rubin CT and Lanyon LE (1984)Dynamic strain similarity in vertebrates; An alternative to allometric limb bone scaling. J. Theor. Biol. 107t321-327. Ruff CB (1987a) Structural allometry of the femur and tibia in Hominoidea and Mucacu. Folia Primatol. (Basel)48t9-49. Ruff CB (1987b3 Sexual dimorphism in human lower limb bone structure: Relationship to subsistence strategy and sexual division of labor. J . Hum. Evol. 16t396416. Ruff CB (1989) New approaches to structural evolution of limb bones in primates. Folia Primatol. (Basell 53:142-159. Ruff CB, Walker AC, and Teaford MF (1989)Body mass, sexual dimorohism and femoral orooortions of Proconsul from Ru‘singa and Mfangino ilands, Kenya. J . Hum. Evol. 18515-536. Schaffler MT, Burr DB, Jungers WL, and Ruff CB (1985) Structural and mechanical indicators of limb specialization in primates. Folia Primatol. (Basel)45%-75. Scott KM (1985) Allometric trends and locomotor adantations in the bovidae. Bull. Am. Mus. Nat. Hi&, 179:197-288. Selker F and Carter DR (1989) Scaling of long bone fracture strength with animal mas: J . BiGmech. 22r1175-1183. Simons EL (19881 A new soecies of Prouithecus (Primates) from Northeast M’adagascar. Fblia Primatol. (Basel)50:143-151. Stahl WRandGumersonJY(1967)Systematic allometry in five species of adult primates. Growth 31:21-34. Stephan H and Bauchot R (1965) Hirn-Korpergewichtsbeziehungen bei den Halbaffen (Prosimii).Acta Zool. 46:209-231. Tattersall I (1982) The Primates of Madagascar. New York: Columbia University Press. Tattersall I (1985)Systematics of Malagasy strepsirhine primates. In DRSwindler and J Erwin (eds.):Comparative Primate Biology 1. New York: Alan R. Liss, Inc. pp. 43-72. Trotter M and Peterson RR (1967)Transverse diameter of the femur: On roentgenograms and on bones. Clin. Orthop. 52:233-239. Yablokov YA (1974)Variability of Mammals. New Dehli: Amerind Publishing Co. (translated from the 1966 Russian edition).
Body Size, Locomotion, and Long Bone Cross-Sectional Geometry in lndriid Primates B DEMES, W L JUNGERS, AND K SELPIEN Department of Anatomical Sciences, State University of New York, Stony Brook, New York 11794 (B D , W L J I and Abteilung Funktionelle Morphologie, Ruhr Uniuersitat, 4630 Bochum, Germany ( K S )
KEY WORDS
Indriids, Long bone geometry, Scaling
ABSTRACT
The geometry of the midshaft cross-sections of the femur and humerus of five indriid species was analysed. Internal (marrow cavity) and external diameters were measured on X-rays in the anteroposterior (a-p) and mediolateral (m-1)planes; cross-sectional areas, second moments of area, and section moduli were calculated using formulae for a hollow ellipse. Cortical thickness, robusticity indices (relating external diameters to the length of the bones), and a-p/m-1 shape variables were also calculated. Model I1 regression was supplemented by analyses of correlation between size and shape. Indriids are saltatory, i.e., their locomotion is dominated by the hind limbs. Accordingly, the femur is more rigid than the humerus, and it shows a consistent difference between the a-p and m-1 planes in measures related to bending strength. Cortical thickness varies considerably both within and across species. The type specimen of the new species Propithecus tattersalli is virtually indistinguishable from P. uerrealwci on the basis of its long bone cross-sectional geometry. Femoral robusticity is uncorrelated with size, but humeral robusticity decreases significantly with increasing size. Femoral shape variables (a-p/m-1) are all negatively correlated with body size, indicating that m-1 dimensions of the femur increase at a faster rate than do a-p dimensions. The highly loaded plane of movement seems to be more reinforced in the smaller species. Contrary to static biomechanical scaling predictions of positive allometry, all cross-sectional parameters scale relatively close to isometry. It is concluded that either changes in locomotor performance must compensate for the weight-related increase in forces and moments or that the larger-bodied animals operate appreciably closer to the limits of their safety margins.
The magnitude and direction of external forces acting on a n animal's bones are determined by its positional repertoire and its weight.' The cross-sectional dimensions of the bones largely determine their ability to withstand the stresses created by these forces. High correlations between locomotion, body size, and cross-sectional dimensions of long bones can therefore be expected. In acknowledging this causal association, an increasing number of studies have collected valuable information on the geometry of long bone cross sections. For various pri'Weight and mass are related by the gravitational constant of 9.81 and may be used interchangeably for animals moving under the influence of earths gravity.
0 1991 WILEY-LISS,INC.
mate species a correlation has been established between the ratio of forelimb to hind limb bone rigidity and the role of the limbs in support and propulsion (Schaffler et al., 1985; Ruff, 1987a; Demes and Jungers, 19891, between cross-sectional shape and locomotor activity (Preuschoft, 1969; Jungers and Minns, 1979; Burr et al., 1982, 1989; Ruff, 1987b, 1989; Demes and Jungers, 1989), and between body weight and crosssectional dimensions, allowing the prediction of weight from cross-sectional data in fossils (Ruff, 1987a, 1989; Ruff et al., 1989). In this study we extend this biomechanical Received December 12,1990; accepted June 19,1991
538
B. DEMES ET AL.
approach to the analysis of skeletal design in to a broad sampling strategy that combines a group of closely related, saltatory pri- animals of different locomotor mode and mates. Leapers have a clear differentiation with different phylogenetic affinities. In a n between the role of the forelimb and hind analysis limited to artiodactyls, but ranging limb in locomotion a s well as a preferentially over three orders of magnitude in body mass, loaded plane for the propulsive hind limb; Selker and Carter (1989) report that crossaccordingly, they should exhibit a pro- sectional parameters scale in a positively nounced functional differentiation in cross- allometric fashion. sectional dimensions. For the present analysis we have deterA related and much debated issue con- mined the midshaft cross-sectional dimencerns the scaling of long bone dimensions sions of the humerus and femur in a group of with body weight. Mechanically inspired closely related prosimian primates in the scaling models predict relative increases family Indriidae.2 The difference in body (positive allometry) in external diameters of mass between the smallest and the largest long bones with increasing body weight. Ga- species in our sample is approximately a lilei (1637) suggested this was due to weight factor of 7 to 8 (see Table 1 and below). In increasing in proportion to volume, and bone analysing closely related species that also cross section increasing in proportion only to share the same general locomotor mode, difarea. Whereas Galilei’s model assumes (im- ferences due to phylogenetic “noise” and plicitly) that axial compression is the pre- variation in locomotor category are reduced dominant loading regime, more recent mod- greatly, thereby revealing size effects more els assume either Euler buckling (McMahon, clearly. This group corresponds closely to 1973; Alexander, 1988) or bending (Kum- what McMahon (1984) h a s recently demer, 1975; Prange, 1977) to be the critical scribed a s the most appropriate type of samloading regimes of long bones. Depending on ple for testing mechanical models of long the model, the expected scaling coefficients bone scaling. By analysing cross-sectional vary, but they all predict values greater than dimensions in addition to external diameisometry or geometrical scaling; i.e., they ters, a much more direct measure of the predict that larger animals should have rel- mechanical strength of a bone is obtained. The five indriid species included in our atively thicker, more robust bones. These scaling models are static in nature and ne- sample are all specialized “vertical clingers glect to account for possible changes in the and leapers” (Napier and Walker, 1967). loading patterns due to differences in loco- They prefer vertical tree trunks for takeoffs motor performance, external forces and their and landings, and their positional repertoire lever lengths, and muscle forces and their is characterized by vertical clinging poslever lengths. Such changes may occur as tures. Quantitative data on locomotor becorrelates of body size (Biewener, 1982, havior are available for sifakas (genus Pro1983,1990; Rubin and Lanyon, 1984; Alex- pithecus) only. About 50% of all locomotor ander, 1985,1989). and postural bouts is leaping, followed by Positive allometry of long bone diameters climbing and clinging (Gebo, 1987; Dagosto, was documented for bovids (McMahon, 1975; 1989).All these activities are hind-limb domAlexander, 1977; Scott, 1985) and for run- inated, which means that the greater share ning birds (Maloiy et al., 1979) and was also of total body weight is supported and accelfound in mixed samples of primate species erated by the hind limbs. Sifakas can easily (Stahl and Gumerson, 1967; Ruff, 1987a). In cover a horizontal distance of 10 meters in a analysing a wide range of different animals single leap (Petter, 1962). The locomotion of from shrews to elephant, Alexander et al. the other species of the family is reported to (1979) and Biewener (1983) found scaling be similar to that of the sifakas (Petter, 1962; coefficients for external diameters closer to Petter et al., 1977; Tattersall, 1982). isometry, but the cross-sectional areas and MATERIALS AND METHODS area moments scale slightly positively alloCross-sectional dimensions were determetrically in ten species of mammals from mouse to horse (Biewener, 1982). Positive mined from measurements taken on radioallometry of cross-sectional areas and area graphs, and a n elliptical model was emmoments was also found in two taxonomically mixed primate samples (Schaffler 2Jenkins (1987) has argued recently that the proper familial et al., 1985; Ruff, 1987a).The interpretation designation for this group is Indridae. We have opted for the “conventional” spelling. of these results is complicated somewhat due
539
INDRIID LONG BONE GEOMETRY
TABLE I. Sample composition Origin
Zndri indri Propithecus diadema P. uerreauxi P. tattersalli Auahi laniger
N
Wild
14/15 10 16/18 1 11/12
7/8 8 9 1 8/9
zoo
?
Body mass
2
7 2 5/7
-
3
7-5 kg (1) 6.2 k g (2,3) 3.7 k g (4,5) 3.2 k g (5) 1.0 kg (1,3,5)
?indicates no locality information; Ns = femur/humerus. if different in numbers. Body weights are basedonthefollowingsources:1:Stephan and Bauchot(l965)report a body weightof6.25 kgfor a wild-shothdrifemale. Judging from its head and body length (Rehkamper, personal communication),this animal is smaller than the species mean (Tattersall, 1985). We have thereforeinserted a higher valueandin addition calculate regressions with two body weights for Zndri;2:Smithsonian Field Collection; 3: Glander et al., 1990; 4: Tattersall, 1985; 5: Duke Primate Center records, Kappler, 1990.
ployed in all calculations. Choice of this simple model seems reasonable because the cross-sectional perimeters of the indriid bones are quite regular, lacking crests and other local protuberances in the midshaft region. For a sample of human humeral sections, Fresia et al. (1990) have recently demonstrated a close correlation between the cross-sectional geometrical properties calculated from measurements taken on X-rays in two planes (using a n elliptical model) and determined directly, lending support to the utility of our approach. The midshaft crosssection was chosen as the focal point of our analyses because there are empirical data indicating that it suffers the highest bending stresses (Biewener and Taylor, 1986). Humeri and femora of five indriid species were X-rayed from the following collections: American Museum of Natural History (New York), Museum National d’Histoire Naturelle (Paris), Anthropologisches Institut und Museum (Zurich), Cleveland Museum of Natural History (Ohio),Museum of Comparative Zoology (Boston), and the Rijksmuseum van Natuurlijke Historie (Leiden). The sample composition is provided in Table 1. Samples include several zoo specimens and animals without precise locality data in order to maximize sample sizes of these relatively rare species. The new species Propithecus tattersalli (Simons, 1988) is represented by the type specimen, so far the only skeleton available for study. All specimens were adult a s judged from their external appearance (i.e., epiphyseal lines unvisible). Individual body masses were not available for the majority of specimens. For the analysis of sizerelated differences, a n average body mass for each species was used (Table 1).Sexes were pooled as all five species are nondimorphic with respect to their body size (Kappler,
1990; Glander et al., 1990). For P. tattersalli, the mass of the type specimen was known from the records of the Duke University Primate Center. The functional length of the bones was measured directly as the greatest distance between proximal and distal joint surfaces. This length measurement may deviate slightly from the greatest length in the case of the femur, where the tip of the greater trochanter can be the most proximal extension. External and internal diameters a t midshaft were measured on the X-rays with a magnifying lens ( x 8 ) with a scale. These measurements were taken in the anteroposterior (a-p) and the mediolateral (m-1) planes, with the planes being determined by the orientation of the distal joint. The a-p plane thus coincides with the plane of movement a t the knee or the elbow joint, respectively, with the m-1 plane being perpendicular to it. The reliability ofthe method was tested for a subset of data by comparing the measurements of external diameters taken from the X-rays with measurements taken on the bone directly with digital calipers. The percentage deviation of the X-ray measurements from the bone measurements was 2% on average. Trotter and Peterson (1967), by comparison, found a 5% deviation for the transverse diameter of the human femur when comparing X-ray with direct bone measurements. Cross-sectional areas (A), a s a measure related to the compressive strength of the bone, and second moments of area (I) as well a s section moduli (21, as reflections of its bending strength, were estimated using formulae for a hollow ellipse (Fig. 1;Roark and Young, 1976):
540
B. DEMES ET AL.
A
= TT
(ab
-
aibJ mm2
L - p - d 4 (a3b - aI3bb,) mm4 Im., = T T / ~(ab3 - a1b,3)mm4
Za.p = IJa mm3 Zm., = ImJb mm3 Ia.p is proportional to the bending rigidity of the cross section against bending moments in the sagittal plane (around a transverse axis); I,,., to the rigidity against bending moments in the frontal plane (around a sagittal axis). The section moduli are a more direct measure of bending strength, because they compensate for the distance from the neutral axis to the bone’s outer perimeter (where the highest bending stresses occur). Both I and Z are reported here because comparative data on the bending strength of long bones are currently available primarily as area moments. The equations used above account for differences in diameters as well as in wall thicknesses between the two planes, but not in wall thicknesses within a plane. As opposite cortices are similar in thickness in our sample, the potential error introduced by this is minor (see Biknevicius and Ruff, in press, for the asymmetrical ellipse model calculations). As a measure of cortical thickness, K-values were determined as internal (marrow cavity) diameter as a ratio of external diameter (Currey and Alexander, 1985). The robusticity index was calculated as the ratio of average external midshaft diameter and bone length. Ratios of diameters, area moments, and section moduli in the two planes (a-p and m-1) were also calculated a s indicators of midshaft cross-sectional shape. The measured and derived variables were compared within each bone for the two planes (a-p and m-11, and within each species for forelimb and hind limb bones (femur and humerus) using the Mann-Whitney U-test; a nonparametric test was employed because variables tended to be nonnormally distributed. The scaling relationship of variables to body size (average body mass) was explored via reduced major axis regressions of logarithmically transformed data (Ricker, 1984; Rayner, 1985). Scaling was also investigated via correlation between size and shape variables (Mosimann and James, 1979). Spearman rank order correlations were calculated between body size and the robusticity indices and a-p/m-1 indices. No significant correlation here implies “isometry”; allometry is
antero posterior
-
mediolateral
Fig. 1. Ellipse approximating cross-sectional areas from anteroposterior and inediolateral diameters. d a , = ?/A externayinternal a-p diameter; b/b, = ‘/z externayinternal m-1 diameter.
indicated simply by significant positive or negative correlations. RESULTS
Variability The coefficients of variation for the linear measurements on the external surface of the bones (length, external midshaft diameters) range between 4.7 and 10.4. This is slightly higher than the variability found in external bone measurements of mammals in general (Yablokov, 1974).This may be due to the fact that some of the species (especially P. uerreauxi and P. diadema) in our sample combine several subspecies with a relatively wide range of body sizes (Tattersall, 1982). For the internal diameter (width of marrow cavity) the coefficients of variation are even higher (6.9-17.61, which is probably due to a somewhat higher measuring error, related to its absolutely smaller dimension and the less clearly demarcated border of the internal surface of the cortical wall. The average coefficient of variation for the cross-sectional areas is 16.1(range: 9.9-24.6), for the section moduli 20.9 (range: 13.5-29.7) and for the second moments of area 28.5 (range: 17.9-
2. p, 2, I = section moduli; I.
Zndri indri femur humerus Propithecus diadema femur humerus Propithecus verreauxi femur humerus Propithecus tattersalli femur humerus Avahi laniger femur humerus 8.7 7.0
9.9 7.8 6.4 f 0.4 4.7 f 0.5
184.9 100.2
128.4 f 7.6 62.3 f 3.3
I, I = second moments of area.
*
6.0 f 0.3 4.9 0.3
+
19.9 i 2.8 9.5 1.5
45.2 21.0
42.8 i 10.5 20.9 f 4.4
9.1 0.9 6.9 i 0.7
9.6 i 0.6 7.2 f 0.7
181.3 f 9.0 97.5 f 6.7
*
69.9 f 6.9 33.2 i 4.2
10.8 f 0.6 8.5 f 0.4
11.3 f 0.7 8.5 f 0.8
209.2 11.4 120.6 f 7.0
+
77.9 f 9.6 36.9 f 6.2
11.7 f 0.7 9.2 f 0.8
Area mm2
+
m-l diam. mm
12.1 f 0.7 9.2 0.8
a-p diam. mm
of
240.2 f 11.7 133.0 f 6.3
mm
Length
TABLE 2. External dimensions and cross-sectional parameters
21.6 f 3.7 8.3 f 2.1
74.8 30.0
71.8 i 17.9 27.4 f 8.1
127.1 f 19.8 50.0 i 9.8
152.7 f 23.0 62.0 f 14.3
mm3
&.&I
L-p
70.0 f 15.7 20.2 f 6.9
19.8 3.1 8.8 k 2.0
+
370.4 117.1
350.1 f 111.6 101.0 f 40.0
67.1 i 19.6 27.2 f 8.0 65.6 28.5
726.3 f 161.8 217.3 f 61.7
928.8 f 188.0 290.3 f 88.9
mm4
119.0 f 16.1 50.1 f 6.8
147.6 f 24.2 61.8 f 14.7
mm3
Zm.1
limb bones (means and one standard deviation) Im-I
59.8 5 12.3 22.0 f 6.7
283.5 99.8
313.0 i 127.1 96.6 f 40.6
+
646.0 f 116.7 213.3 38.3
868.4 k 195.9 290.1 f 93.9
mm4
542
B. DEMES ET AL
tion of two K-values in the mediolateral plane (P. diadema: P < 0.05; P. uerreauxi: n.s.1. Relative to its length the humerus is, however, the more robust bone (Table 4).
T A B L E 3. Ratios of internal to external midshaft diameters (K-values) (means and one standard deviation)
Indri indri femur humerus Propithecus diadema femur humerus Propithecus uerreauxi femur humerus Propithecus tattersalli femur humerus Auahi laniger femur humerus K
=
K-value a-p
K-value m-l
0.55 f 0.03 0.66 f 0.05
0.54 f 0.05 0.67 f 0.05
0.49 k 0.06 0.65 f 0.05
0.54 f 0.08 0.63 f 0.06
0.60 f 0.07 0.70 f 0.04
0.63 f 0.07 0.66 f 0.04
0.57 0.74
0.59 0.68
0.56 f 0.05 0.70 f 0.04
0.60 f 0.04 0.68 f 0.04
internal diametedexternal diameter
42.1). In much larger samples of hominoid primates, Ruff (1987a) reports similarly high coefficients of variation (7-26 for crosssectional areas, and 14-47 for polar moments of inertia). These data demonstrate that this degree of variability is not due to small sample sizes; rather, it is simply the consequence of higher than linear dimensionality such a s mm2 and mm4 (Lande, 1977).
Femur us. humerus The femur has greater outer diameters, wall thicknesses, areas, second moments of area, and section moduli than does the humerus (Tables 2, 3). All differences are significant a t the P < 0.01 level, with the excep-
a-p us. m-1plane In the femur, there is a consistent difference between the a-p and the m-1 plane such that the a-p diameters, cortical thicknesses, second moments of area, and section moduli are greater than the corresponding m-1 parameters (with one exception of wall thickness in Indri; Tables 2 4 ) . However, none of these differences is statistically significant a t the P < 0.01 level. There are no consistent differences between the two planes in the humerus; i.e., a-p and m-1 variables are very similar (Tables 2 4 ) . K-values The wall thicknesses vary considerably, within a cross section between the a-p and m-1 plane, between the femur and humerus of the same species, and interspecifically. Figure 2 illustrates this variation for the femur by contrasting a n a-p/m-1plot of K-values with one for section moduli. The femora tend to have lower K-values (i.e., thicker cortical walls) than do the humeri (Table 3). The variation in K does not show any obvious correlation with size, taxonomy, or with the two planes of a bone. Size All variables are highly correlated with body mass (Table 5). With the exception of bone length (cf. Jungers, 19851, most scaling coefficients are quite close to isometry: for
T A B L E 4. Robusticity and diaphyseal shape of limb bones (means and one standard deviation)
Zndri indri femur humerus Propithecus diadema femur humerus Propithecus uerreauxi femur humerus Propithecus tattersalli femur humerus Auahi laniger femur humerus Robusticity index = (a-p diameter
Robusticity index
a-p diam. m-1 diam.
1a.p
2a.p
1m.I
Zm.1
9.9 f 0.7 13.9 f 1.0
1.039 i 0.070 1.002 f 0.084
1.085 i 0.149 1.010 f 0.163
1.040 f 0.080 1.004 f 0.071
10.6 i 0.7 14.1 f 0.9
1.049 +_ 0.052 1.008 f 0.076
1.123 i0.119 1.006 -+ 0.164
1.067 k 0.064 0.992 f 0.095
10.3 f 0.7 14.5 f 0.9
1.062 f 0.069 1.042 f 0.087
1.151 f 0.149 1.055 f 0.159
1.080 f 0.073 1.007 f 0.070
10.1 14.8
1.140 1.013
1.306 1.173
1.140 1.053
9.7 f 0.5 15.5 f 1.3
1.074 f 0.055 0.964 f 0.054
1.175 f 0.130 0.914 0.108
1.091 f 0.067 0.945 f 0.060
+ m-l diameter) X 100Aength. Ia.p, I,
1=
*
second moments of area; Z.
p,
Z,.
= section moduli.
543
INDRIID LONG BONE GEOMETRY
O8
r250 200
l------
E
0.5
T Avahi loniger
V Propithecus verreaux 0
P. tattersaiii
0 P. diaderna 0 indri indri
0.4 0.4
0.5
k-value
0.8
0.7
0.6 0-D
femur
0
50
100 0-p
150
:
200
section mod. f e m u r
Fig. 2. Femur m-1 parameters plotted against corresponding a-p parameters. a: K-values; b: section moduli.
TABLE 5. Reduced major axis regressions of variables against body mass (log-log transformed data) Bone length femur humerus Diameter a-p femur humerus Diameter m-1 femur humerus Cross-sectional area femur humerus Section modulus a-p femur humerus Section modulus m-1 femur humerus Area moment a-p femur humerus Area moment m-1 femur humerus
Correlation
y-intercept
Slope
0.983 0.993
4.844 4.130
0.301 (0.290) 0.377 (0.363)
0.992 0.978
1.868 1.569
0.319 (0.307) 0.336 (0.323)
0.998 0.991
1.781 1.574
0.335 (0.322) 0.314 (0.302)
0.992 0.992
2.964 2.225
0.696 (0.669) 0.692 (0.665)
0.996 0.993
3.075 2.117
0.987 (0.949) 1.008 (0.969)
0.997 0.994
2.971 2.149
1.009 (0.970) 0.981 (0.943)
0.995 0.992
4.260 3.016
1.305 (1.255) 1.339 (1.287)
0.998 0.994
4.066 3.046
1.344 (1.292) 1.292 (1.242)
Correlations, intercepts, and slopes are given forregressions calculated with a body mass of 7 kgfor Indri.They do not changemuch if Indri is inserted with a body mass of 8 kg (slopes in parentheses; see Methods section).
the linear dimensions (diameters) approximately 0.33 W31, for areas approximately 0.67 (2/3), for section moduli approximately 1 ( 3 / 3 ) , and for area moments approximately 1.33 (Y3). Femoral robusticity follows this isometry trend in that this index is uncorrelated with body mass (Table 6, Fig. 3). Hum-
era1 robusticity, however, decreases significantly with increasing body mass (Table 6). Zndri has the least robust humerus and Avahi has the most robust (Table 4, Fig. 3); presumably, the positive allometry of humerus length contributes to this trend (note that these data modify the results reported
544
B. DEMES ET AL.
TABLE 6. Rank-order correlations between body mass and diaphyseal shape variables Variable Humerus robusticity index a-p/m-l diameter Ia-p/Im-l
Za.p/Zm.l
Femur robusticity index a-p/m-1 diameter Ia.p~Im.l Z,./Z, I
Spearman correlation
Interpretation
-1.000* 0.100 0.100 0.100
neg. allometry isometry isometry isometry
0.400 -0.900* -0.900* -0.900*
isometry neg. allometry neg. allometry nee. allometrv
*Indicates significance (P< 0.05); negative allometry implies that index decreases; isometry implies no significant change in index with increasing size (geometric similarity preserved).
by Jungers, 1985, for the humerus of indriids). Significant deviations from isometry are also reflected in shape changes of the femoral midshaft cross-section with increasing size. The a-p to m-1 ratios of diameters, second moments of area, and section moduli decrease with increasing size (Table 61, indicating that the difference in strength between these two planes is less pronounced in the larger-bodied species. These significant size-related shape changes are the combined effects of the slightly divergent a-p and m-1 scaling trends seen in Table 5. DISCUSSION
Our data corroborate hypotheses about the functional differentiation of long bones according to habitual positional behaviors. The bone of the propulsive hind limb (femur) that has to withstand the higher forces is more resistant to loads than is the bone ofthe corresponding segment of the forelimb (humerus). The femur also offers the greater resistance in the plane of movement (a-p), where it is probably exposed to higher bending moments than in the plane perpendicular to it (m-1). This difference is, however, quite subtle, and changes with size, so that inferences derived solely from cross-sectional shape about the likely mode of locomotion in fossils should be stated carefully. It should also be mentioned in this context that there are examples in which the long axis of the elliptical cross section of a bone is not in the plane of major bending (Lanyon and Rubin, 1985). In some experiments changing or increasing the loading of bones, on the other hand, bone apposition and mechanical reinforcement is found in the loaded plane
(e.g., Liskova & Helt, 1971; Gordon, et al., 1989). The external measurements distinguish the two bones and the two planes within a bone in much the same way a s do the crosssectional parameters. They accurately predict aspects of internal geometry of the bones and can therefore be taken a s indirect indicators of their mechanical strength when cross-sectional data cannot be obtained (Ruff, 1987b). A close correlation between external bone diameters and second moments of cross-sectional area was also documented by Jungers and Minns (1979) for a mixed primate sample. This is not totally unexpected as the outer parts of cross-sectional area contribute preferentially to bending strength (compare equations for I and Z in the Methods section). The one specimen of the new species P. tattersalli, although weighing slightly less than a n average P. uerreauxi, falls in the upper range of verreauxi’s spectrum with regard to its humeral and femoral dimensions (Tables 2-4). The type specimen is a very robust male. The variation in cortical thickness does not exhibit a very coherent pattern. K-values for the femur vary around an optimal value of 0.55, predicted by Currey and Alexander (1985) to achieve maximum ultimate bending strength with a minimum of bone mass. K-values for the humerus lie closer to what is the assumed optimum for yield bending strength (0.65:Pauwels, 1974; 0.67: Currey and Alexander, 1985). However, both femoral and humeral K-values vary considerably. These data, as well as similarly variable data on lorisine long bones (Demes and Jungers, 19891, suggest that achieving a maximum bending strength with a minimum of bone mass is probably not the only factor determining the wall thickness of long bones. The virtual isometry of all cross-sectional parameters with size clearly contradicts all competing static scaling predictions of positive allometry (Galilei, 1637; McMahon, 1973; Kummer, 1975; Prange, 1977; Alexander, 1988). The findings also suggest that Biewener’s (1990) size categories (e.g., isometry limited to mammals between 0.001-0.1 kg) are not hard and fast. In addition, they stress the importance of multifactorial approaches toward bone scaling, taking into account changes in locomotor performance and external forces with size (Biewener, 1983, 1990; Rubin and Lanyon, 1984: dynamic strain similarity; Alexander, 1989).
INDRIID LONG BONE GEOMETRY
545
G
a U
Fig. 3. Femur and humerus of Auahi laniger (right) and Indri indri (left)drawn at the same length (scale bars indicate 1 cm). Notice the similar femur proportions (a)and the increased humeral robusticity of the smaller-bodied species (b).
Options to cope with increasing weight-related forces include changes in locomotor behavior that reduce the external loads (e.g., reduce speed; Alexander, 1989) and their moment arms (e.g., straighter legs, shorter strides; Alexander, 1989; Biewener, 1989, 1990; Bertram and Biewener, 1990), and changes in body proportions (e.g., reduce bone length and bony moment arms, increase muscle moment arms; Biewener, 1990). For example, leaping galagos and L. catta show distinct, size-related differences in the way they accelerate for takeoff. The small-bodied species exert relatively higher forces and assume smaller angles a t the joints of the accelerating hind limb than do the larger-bodied species (Gunther, 1989; Demes and Gunther, 1989). Although corresponding data for the indriids are not yet available, it is not unlikely that they compensate in a similar way for increasing weight-related forces. The decreasing a-p to m-1 ratios in variables of the femur cross section related to bending strength support this view. Alternatively, it has to be assumed
that the bones of the larger-bodied animals suffer from greater stresses and operate closer to their safety margins (Biewener, 1982). In vivo strain gage data, however, indicate, that bone strains, although variable, are in the same order of magnitude in animals over a wide range of body sizes (Rubin and Lanyon, 1984). When comparing our results with those of other studies of long bone scaling, it is obvious that animals follow different strategies to maintain strain similarity with increasing size (although dynamic similarity seems to apply a t least to a certain degree: Alexander, 1985). In other analyses of closely related species that also share the same mode of locomotion, slight to moderate positive allometry was found for most of the crosssectional parameters of artiodactyl long bones and for “pongid” tibiae and femora (Ruff, 1987a1, and strong positive allometry for lorisine long bones (Demes and Jungers, 1989). I n lorises, the increase in bone dimensions with increasing size is even more pronounced than could be expected from docu-
546
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mented changes in kinematic and dynamic parameters with increasing size (Demes et al., 1990), raising the likelihood of greater safety factors for the larger-bodied species as a n adaptation to infrequent but critical stresses (Alexander, 1984). Indriids appear to have “opted” for a rather different solution given the virtual isometry in their long bone cross-sectional dimensions. We still have much to learn about the interactions in body size, skeletal design, and animal behavior, but we believe that the analytical strategy used here that focuses on closely related animals of different size but similar design has much to offer. ACKNOWLEDGMENTS
We would like to thank the curators of the collections listed under “Methods” for giving us access to specimens in their care. Special thanks for help with the radiographs go to A. Biknevicius and N. Feinberg (New York), F.A. Jenkins (Harvard), F.K. Jouffroy (Paris), B. Latimer (Cleveland), P. Schmid (Zurich), C. Smeenk and S. Zeilstra (Leiden). Unpublished body size data were kindly provided to us by B. Colfman (Duke University Primate Center) and G. Rehkamper (Cologne). This research was supported in part by DFG grant De 390/1-1 and NSF BNS 8606781. LITERATURE CITED Alexander RMcN (1977) Allometry of the limbs of antelopes (Bouidae).J. Zool., Lond. 183t125-146. Alexander RMcN (1984) Optimum strength for banes liable to fatigue and accidental fracture. J. Theor. Biol. 109:621-636. Alexander RMcN (1985) Body support, scaling, and allometry. In M Hildebrand, DM Bramble, KF Liem, and DB Wake (eds.):Functional Vertebrate Morphology. Cambridge, Mass.: The Belknap Press, pp. 26-37. Alexander RMcN (1988) Elastic Mechanisms in Animal Movement. Cambridge: Cambridge University Press. Alexander RMcN (1989)Mechanics of fossil vertebrates. J . Geol. SOC., Lond. 146:41-52. Alexander RMcN, Jayes AS, Maloiy GMO, and Wathuta EM (1979) Allometry of the limb bones of mammals from shrews (Sorex)to elephant (Loxodonta).J. Zool., Land. 189:305-3 14. Bertram JEA and Biewener AA (1990)Differential scaling of the long bones in the terrestrial Carnivora and other mammals. J. Morphol. 204t157-169. Biewener AA (1982) Bone strength in small mammals and bipedal birds: Do safety factors change with body size? J. Exp. Biol. 98t289-301. Biewener AA (1983) Allometry of quadrupedal locomotion: The scaling of duty factor, bone curvature and limb orientation to body size. J. Exp. Biol. 105t147171. Biewener AA (1989) Mammalian terrestrial locomotion and size. BioScience 139:776-783.
Biewener AA (1990)Biomechanics of mammalian terrestrial locomotion. Science 250:1097-1103. Biewener AA and Taylor CR (1986) Bone strain: A determinant of gait and speed? J. Exp. Biol. 123:383400. Biknevicius AR and Ruff CB (in press) Use of biplanar radiographs for estimating cross-sectional geometric properties of mandibles. Anat. Rec. Burr DB, Piotrowski G, Martin BR, and Cook PN (1982) Femoral mechanics in the lesser bushbaby (Galago senegalensis): Structural adaptations to leaping in primates. Anat. Rec. 202:419429. Burr DB, Ruff CB, and Johnson C (1989) Structural adaptation of the femur and humerus to arboreal and terrestrial environment in three species of macaque. Am. J. Phys. Anthropol. 79t357-367. Currey J D and Alexander RMcN (1985)The thickness of the walls oftubular bones. J. Zool., Lond. 206:453-468. Dagosto M (1989) Locomotion of Propithecus diadema and Vurecia uarzegata at Ranomafana National Park, Madagascar. Am. J. Phys. Anthropol. 78t209. Demes B and Gunther MM (1989) Biomechanics and allometric scaling in primate locomotion and morphology. Folia Primatol. (Basel)53t125-141. Demes B and Jungers WL (1989)Functional differentiation of long bones in lorises. Folia Primatol. (Basel) 52t58-69. Demes B, Jungers WL, and Nieschalk U (1990)Size- and speed-related aspects of quadrupedal walking in slender and slow lorises. In FK Jouffroy, MH Stack, and C Niemitz (eds.): Gravity, Posture and Locomotion in Primates. Florence: I1 Sedicesimo, pp. 175-197. Fresia AE, Ruff CB, and Larsen CS (1990) Temporal decline in bilateral asymmetry of the upper limb on the Georgia coast. In CS Larsen (ed.):The Archaeology of Mission Santa Catalina de Guale: 2. Biocultural Interpretations of a Population in Transition. New York: Anthropological Papers of the American Museum of Natural History 68:121-150. Galilei G (1637)Discorsi e Dimostrazioni Mathematiche Intorno a Due Nuove Scienze. New, translated edition (1914).Dialogues Concerning two New Sciences. New York: MacMillan. Gebo DL (1987) Locomotor diversity in prosimian primates. Am. J. Primatol. 13:271-281. Glander KE, Wright PC, and Daniels PS (1990)Morphometrics of six rain forest prosimian species from Southeastern Madagascar. Am. J. Primatol. 81229. Gordon KR, Per1 M, and Levy C (1989) Structural alterations and breaking strength of mouse femora exposed to three activity regimens. Bone 10:303-312. Gunther MM (1989)Funktionsmorphologische Untersuchungen zum Sprungverhalten an mehreren Halbaffenarten. Berlin: Ph.D. thesis. Jenkins PD (1987) Catalogue of Primates in the British Museum of Natural History. Part IV. Suborder Strepsirrhini. London: British Museum. Jungers WL (1985)Body size and scaling of limb proportions in primates. In WL Jungers (ed.): Size and Scaling in Primate Biology. New York: Plenum Press, pp. 345-381. Jungers WL and Minns R J (1979) Computed tomography and biomechanical analysis of fossil long bones. Am. J. Phys. Anthropol. 50:285-290. Kappler PM (1990) The evolution of sexual dimorphism in prosimian primates. Am. J. Primatol. 21:201-214. Kummer B (1975) Grundsatzliche Bemerkungen zum Einfluss der Korpergrosse und der Gravitation auf die Konstruktion des Bewegungsapparates landbe-
INDRIID LONG BONE GEOMETRY wohnender Tetrapoden. Aufsatze u. Reden Senckenb. naturf. Ges. 27:69-84. Lande R (1977) On comparing coefficients of variation. Syst. Zool. 26:214-217. Lanyon LE and Rubin CT (1985) Functional adaptation in skeletal structures. In M Hildebrand, DM Bramble, KF Liem, and DB Wake (eds.): Functional Vertebrate Morphology. Cambridge, Mass.: The Belknap Press, pp. 1-25. Liskova M and Heit J (1971)Reaction of bone to mechanical stimuli. Part 2. Periosteal and endosteal reaction of tibia1 diaphysis in rabbit to intermittent loading. Folia Morphol. (Warsz) 19:301-317. Maloiv GMO. Alexander RMcN. Niau R. and Javes AS (19f9)Allometry of the legs of rtnning birds. i.Zool., Lond. 187:161-167. McMahon TA (1973) Size and shape in biology. Science 179:1201-1204. McMahon TA (1975) Allometry and biomechanics: Limb bones in adult ungulates. Am. Natur. 109:547-563. McMahon TA (1984)Muscles, Reflexes, and Locomotion. Princeton: Princeton University Press. Mosimann J E and James FC (1979) New statistical methods for allometry with application to Florida red-winged blackbirds. Evolution 33t444459. Napier JR and Walker AC (1967) Vertical clinging and leaping-a newly recognized category of locomotor behaviour of primates. Folia Primatol. (Basel) 6204219. Pauwels F (1974)Uber die Bedeutungder Markhohle fur die mechanische Beanspruchung des Rohrenknochens. Z. Anat. Entwick1.-Gesch. 145t81-85. Petter JJ (1962) Recherches sur l’l&ologie et 1’Ethologie des Lemuriens Malgaches. Mem. Mus. Nat. dHist. Nat. 27(1). Petter JJ, Albignac R, and Rumpler Y (1977) Mammiferes lemuriens (Primates, Prosimiens). Faune Madagascar 44: 1-513. Prange HD (1977) The scaling and mechanics of artbropod exoskeletons. In J T Pedley (ed.): Scale Effects in Animal Locomotion. London: Academic Press, pp. 169-1 81. Preuschoft H (1969) Statische Untersuchungen am Fuss der Primaten. I. Statik der Zehen und des Mittelfusses. Z. Anat. Entw. Gesch. 129:285-345. Rayner JMV (1985) Linear relations in biomechanics: The statistics of scaling functions. J. Zool., Lond. 206:415439. Ricker WE (1984)Computation and uses ofcentral trend lines. Can. J. Zool.62t1897-1905.
547
Roark R J and Young WC (1976)Formulas for Stress and Strain, 5th Edition. Auckland: McGraw-Hill. Rubin CT and Lanyon LE (1984)Dynamic strain similarity in vertebrates; An alternative to allometric limb bone scaling. J. Theor. Biol. 107t321-327. Ruff CB (1987a) Structural allometry of the femur and tibia in Hominoidea and Mucacu. Folia Primatol. (Basel)48t9-49. Ruff CB (1987b3 Sexual dimorphism in human lower limb bone structure: Relationship to subsistence strategy and sexual division of labor. J . Hum. Evol. 16t396416. Ruff CB (1989) New approaches to structural evolution of limb bones in primates. Folia Primatol. (Basell 53:142-159. Ruff CB, Walker AC, and Teaford MF (1989)Body mass, sexual dimorohism and femoral orooortions of Proconsul from Ru‘singa and Mfangino ilands, Kenya. J . Hum. Evol. 18515-536. Schaffler MT, Burr DB, Jungers WL, and Ruff CB (1985) Structural and mechanical indicators of limb specialization in primates. Folia Primatol. (Basel)45%-75. Scott KM (1985) Allometric trends and locomotor adantations in the bovidae. Bull. Am. Mus. Nat. Hi&, 179:197-288. Selker F and Carter DR (1989) Scaling of long bone fracture strength with animal mas: J . BiGmech. 22r1175-1183. Simons EL (19881 A new soecies of Prouithecus (Primates) from Northeast M’adagascar. Fblia Primatol. (Basel)50:143-151. Stahl WRandGumersonJY(1967)Systematic allometry in five species of adult primates. Growth 31:21-34. Stephan H and Bauchot R (1965) Hirn-Korpergewichtsbeziehungen bei den Halbaffen (Prosimii).Acta Zool. 46:209-231. Tattersall I (1982) The Primates of Madagascar. New York: Columbia University Press. Tattersall I (1985)Systematics of Malagasy strepsirhine primates. In DRSwindler and J Erwin (eds.):Comparative Primate Biology 1. New York: Alan R. Liss, Inc. pp. 43-72. Trotter M and Peterson RR (1967)Transverse diameter of the femur: On roentgenograms and on bones. Clin. Orthop. 52:233-239. Yablokov YA (1974)Variability of Mammals. New Dehli: Amerind Publishing Co. (translated from the 1966 Russian edition).
